Problem 61 - College Algebra

Question

How can you distinguish parabolas from other conic sections by looking at their equations?

Step-by-Step Solution

Answer

Parabolas can be distinguished from other conic sections by examining their equations; an equation that includes a single squared variable (either 'x' or 'y') typically represents a parabola.

Step 1: Familiarize with Conic Sections

Conic sections are curves obtained as the intersection of the surface of a cone with a plane. There are four types of conic sections: these include circle, ellipse, parabola, and hyperbola.

Step 2: Understand the Standard Forms

Each conic section has a standard form. The general standard forms are: \n- Circle: \(x^2 + y^2 = r^2\) \n- Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (for horizontal), or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) (for vertical) \n- Parabola: \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\)\n- Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Step 3: Identify a Parabola

A parabola is the set of points that are an equal distance from a point (the focus) and a line (the directrix). In the equation of a parabola, only one variable is squared. This is the distinguishing feature of a parabola. So if you see an equation where only the 'x' or only the 'y' is squared, you are looking at a parabola. Identifying this key feature in the equation will help distinguish a parabola from other conic sections.